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Fund Quantum Mechanics Lect & HW Solutions 44

Fund Quantum Mechanics Lect & HW Solutions 44 - one...

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26 CHAPTER 3. BASIC IDEAS OF QUANTUM MECHANICS Answer: Almost every word in the above story is a gross misstatement of what nature really is like when examined on quantum scales. A particle does not have a position, so phrases like “hits an end”, “reflect backward”, and “keep moving” are truly meaningless. On macroscopic scales a particle may have an relatively precisely defined position, but that is only because there is uncertainty in energy. If you could bring a macroscopic particle truly into a single energy eigenstate, it too would have no position. And the smallest thing you might do to figure out where it is would kick it out of that single energy state. 3.5.8 Three-dimensional solution 3.5.8.1 Solution pipeg-a Question: If the cross section dimensions y and z are one tenth the size of the pipe length, how much bigger are the energies E y 1 and E z 1 compared to E x 1 ? So, by what percentage is the one-dimensional ground state energy E x 1 as an approximation to the three-dimensional
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Unformatted text preview: one, E 111 , then in error? Answer: The energies are E x 1 = ¯ h 2 π 2 2 mℓ 2 x E y 1 = ¯ h 2 π 2 2 mℓ 2 y E z 1 = ¯ h 2 π 2 2 mℓ 2 z . If ℓ y and ℓ z are ten times smaller than ℓ x then E y 1 and E z 1 are each 100 times larger than E x 1 . So the one-dimensional ground state energy E x 1 is smaller than the true ground state energy E 111 = E x 1 + E y 1 + E z 1 by a factor 201. Which means it is o² by 20,000%. 3.5.8.2 Solution pipeg-b Question: At what ratio of ℓ y /ℓ x does the energy E 121 become higher than the energy E 311 ? Answer: Using the given expression for E n x n y n z , E n x n y n z = n 2 x ¯ h 2 π 2 2 mℓ 2 x + n 2 y ¯ h 2 π 2 2 mℓ 2 y + n 2 z ¯ h 2 π 2 2 mℓ 2 z , E 121 = E 311 when ¯ h 2 π 2 2 mℓ 2 x + 4¯ h 2 π 2 2 mℓ 2 y + ¯ h 2 π 2 2 mℓ 2 z = 9¯ h 2 π 2 2 mℓ 2 x + ¯ h 2 π 2 2 mℓ 2 y + ¯ h 2 π 2 2 mℓ 2 z...
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